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Chapter 2: Solving Equations and Inequalities (Page 1 of 40) 2.1 Linear Equations and Problem Solving Example 1 Equations with Fractions x 3x Solve + =2 3 4 Example 2 Solve To simplify an equation containing fractions, first multiply every term in the equation by the LCD of the terms. Then solve the simpler equation. Watch for Extraneous Solutions 1 3 6x = − 2 x−2 x+2 x −4 2.1 Linear Equations and Problem Solving (Page 2 of 40) Mathematical Modeling With Linear Equations The process of translating phrases or sentences into algebraic expressions or equations is called mathematical modeling. A mathematical model is an equation describing a real-life situation. One method of modeling involves two stages: (1) Use the verbal statement of the problem to form a verbal model of the problem; (2) Assign variables to quantities and form the mathematical model (equation). Verbal Description Verbal Model Mathematical Model (equation) Example 1 Using a Verbal Model You accept a job for which the annual salary will be $32,300. This salary includes a year-end bonus of $500. You will be paid twice a month. What will be your gross pay (before taxes) of each paycheck? Example 2 Find the Percent of a Raise You accepted a job that pays $8 an hour. You are told that after a two-month probationary period, you wage will increase to $9 per hour. What percent raise will you earn after the probationary period? 2.1 Linear Equations and Problem Solving (Page 3 of 40) Example 3 Find the Percent Monthly Expenses Your family has an annual income of $57,000 and the following monthly expenses: mortgage ($1100), car payment ($375), food ($295), utilities ($240) and credit cards ($220). What percent of your total family income are your total monthly expenses? Example 4 Dimensions of a Room A rectangular kitchen is twice as long as it is wide, and its perimeter is 84 feet. Find the dimensions of the kitchen. Example 5 A Distance Problem A plane is flying non-stop from Atlanta to Portland, a distance of 2700 miles. After 1.5 hours in the air, the plane flies over Kansas city (a distance of 820 miles from Atlanta). Estimate the time it will take the plane to fly from Atlanta to Portland. 2.1 Linear Equations and Problem Solving (Page 4 of 40) Example 6 Similar Triangles To determine the height of the Aon Center Building (in Chicago), you measure the shadow cast by the building and find it to be 142 feet long. Then you measure the shadow cast by a 4-foot post and find it to be 6 inches long. Estimate the height of the building. Example 7 Simple Interest, I = Prt You invest a total of $10,000 at 4.5% and 5.5% interest. During one year the two accounts earned $508.75. How much did you invest in each account? 2.1 Linear Equations and Problem Solving (Page 5 of 40) Example 8 An Inventory Problem A store has $30,000 of inventory in single-disc DVD players and multi-disc DVD players. The profit on a single-disc player is 22% and the profit on the multi-disc player is 40%. The profit for the entire stock is 35%. How much was invested in each type of DVD player? Example 9 Using a Formula A cylindrical can has a volume of 200 cubic centimeters and a radius of 4 centimeters. Find the height of the can. The volume of a cylinder is V = π r 2 h . See “Common Formulas” on page 170. 2.2 Solve Equations Graphically (Page 6 of 40) 2.2 Solving Equations Graphically Intercepts in a Graph 1. An x-intercept is an ordered pair in the form (x, 0). To find the x-intercepts, set y = 0 and solve for x. 2. A y-intercept is an ordered pair in the form (0, y). To find the y-intercepts, set x = 0 and solve for y. Example 1 Find the Intercepts Find the x- and y-intercepts in the graph of 2x + 5y = 10 . The Zeros of a Function The zeros of a function f are the x-values for which f (x) = 0 . Example 2 Verify the Zeros of a Function a. Verify that the numbers -2 and 3 are zeros of f (x) = x 2 − x − 6 . b. What are the x-intercepts of the graph of f (x) = x 2 − x − 6 . The Zeros and x-Intercepts The following statements are equivalent: 1. The point (a, 0) is an x-intercept in the graph of f. 2. The number a is a zero of f. 2.2 Solve Equations Graphically (Page 7 of 40) 3. The number a is a solution to f (x) = 0 . That is, f (a) = 0 Finding Solutions Graphically Example 3 Find Solutions Graphically Use a graphing utility to approximate the solutions of 2x 2 − 3x = −2 Example 4 Find Solutions Graphically Use a graphing utility to approximate the solutions of x 2 + 3 = 5x 2.2 Solve Equations Graphically (Page 8 of 40) Example 5 Finding Points of Intersection a. Find the points of intersection in the graphs of y = x + 2 and y = x 2 − 2x − 2 . b. Solve x + 2 = x 2 − 2x − 2 Example 6 Finding Points of Intersection Find the points of intersection in the graphs of 2x − 3y = −2 and 4x − y = 6 . Example 7 Finding Points of Intersection a. Find the points of intersection in the graphs of y = x 2 − 3x + 4 and y = x 3 + 3x 2 − 2x − 1. b. Solve x 2 − 3x + 4 = x 3 + 3x 2 − 2x − 1 2.2 Solve Equations Graphically (Page 9 of 40) 2.3 Complex Numbers (Page 10 of 40) 2.3 Complex Numbers So far we have said the equation x 2 = −1 has no real solutions. However, by introducing complex numbers, we can solve such equations. The Imaginary Unit i The imaginary unit, written i, is the number whose square is –1. That is, i 2 = −1 or i = −1 Complex Number a + bi A complex number in standard form is written a + bi, where a and b are real numbers. If b = 0, then a + bi = a is a real number. If b ≠ 0 , then a + bi is an imaginary number. If a = 0 and b ≠ 0 , then a + bi = bi is called a pure imaginary number. Complex Number 2 + 5i − 23 − 43 i 5 Real Part Complex Part Real, Imaginary, or Pure Imaginary a bi −i 2 Example 1 Add & Subtract Complex Numbers (4 + 7i) + (1− 6i) a. b. (1+ 2i) − (4 + 2i) 2.3 Complex Numbers (Page 11 of 40) Example 2 Multiply Complex Numbers −3(5 − 6i) (1+ 2i)(4 + 2i) a. b. c. (3 + 2i)2 d. (3 + 4i)(3 − 4i) Complex Conjugates and Their Product The numbers a + bi and a – bi are complex conjugates of each other. The product of a complex number and its conjugate is a real number, that is, (a + bi)(a − bi) = a 2 + b 2 Example 3 Multiply Complex Conjugate Multiply each complex number by its conjugate. 3+ i 3 − 4i a. b. Example 4 Simplify a Quotient 3 − 4i Simplify . 4 − 2i Fact: A quotient is not simplified until the denominator is a real number. 2.3 Complex Numbers (Page 12 of 40) Square Root of a Negative Number If a is a positive real number, then For example: −a = i a − 4 = −1 4 = i 4 = 2i . Example 5 Simplify each complex number and write in a + bi form. 1. −18 2. − −24 3. −3 −12 5. −6i 3 + i 2 4. (−2 + −3)2 6. (−2i)−3 2.4 Solving Equations Algebraically (Page 13 of 40) 2.4 Solving Quadratic Equations Algebraically A quadratic equation in general form is a second-degree polynomial equation written in the form ax 2 + bx + c = 0 , where a ≠ 0 . Furthermore, ax 2 is called the quadratic term, bx is the linear term and c is the constant term. The methods we study to solve quadratic equations are factoring, extracting square roots, completing the square, and using the Quadratic Formula. The factoring method requires the Zero-Factor Property: If ab = 0 , then a = 0 or b = 0. Example 1 Solve a Quadratic Equation by Factoring 2x 2 + 9x + 7 = 3 a. Solve and check b. Solve and check 6x 2 − 3x = 0 2.4 Solving Equations Algebraically (Page 14 of 40) Extracting Square Roots The equation u 2 = d , where d > 0, has exactly two solutions: u= d and u=− d Example 2 Extracting Square Roots 2 a. Solve 4x = 12 b. Solve 9(x − 3)2 = 28 Completing The Square To complete the square on the expression x 2 + bx , add (b / 2)2 , that is add the square of half the linear coefficient. Then, 2 2 b⎞ ⎛ b⎞ ⎛ 2 x + bx + ⎜ ⎟ = ⎜ x + ⎟ . ⎝ 2⎠ ⎝ 2⎠ Example 3 For each show the constant term of the perfect-square trinomial is one-half of the coefficient of its linear term squared 1. (x + 5)2 = x 2 + 10x + 25 2. (x − 7)2 = x 2 − 14x + 49 Example 4 Find the value of c so the expression is a perfect square trinomial. Then factor each perfect square trinomial. 1. x 2 + 12x + c 2. x 2 − 8x + c 3. x 2 + 4x + c 2.4 Solving Equations Algebraically (Page 15 of 40) Steps To Solve ax 2 + bx + c = 0 by Completing the Square 1. Write the equation in the form ax 2 + bx = − c . That is, get all the variable terms on one side of the equation and the constant term on the other side. 2. The completing the square procedure requires the leading coefficient to be one, so divide both sides by a. Then b c x2 + x = − a a b x a perfect square trinomial, a add half the linear coefficient squared to both sides of the equation. Solve 3x 2 − 18x − 3 = 0 1. 3x 2 − 18x = 3 2. 3x 2 − 18x 3 = 3 3 x 2 − 6x = 1 3. To make x 2 + Since ⎡⎣ 12 ( −6 ) ⎤⎦ = (−3)2 = 9 3. x 2 − 6x + 9 = 1+ 9 x 2 − 6x + 9 = 10 4. Factor the perfect square trinomial: (x + half the linear coefficient)2 4. (x − 3)2 = 10 5. Solve by taking square roots. 5. 2 (x − 3)2 = ± 10 x − 3 = ± 10 6. Check your solutions. x = 3 ± 10 x = 3 ± 10 x ≈ −0.1623, 6.1623 You can also verify the graph of y = 3x 2 − 18x − 3 has x-intercepts at (− 0.162, 0) and (6.162, 0) . 6. 2.4 Solving Equations Algebraically (Page 16 of 40) Example 3 Completing the Square 2 Solve x + 2x − 6 = 0 by completing the square. 1. Write the equation in the form ax 2 + bx = − c . That is, get all the variable terms on one side of the equation and the constant term on the other side. 2. The completing the square procedure requires the leading coefficient to be one, so divide both sides by a. Then b c x2 + x = − a a 3. To make x 2 + 4. Factor the perfect square trinomial: b x a perfect a square trinomial, add half the linear coefficient squared to both sides of the equation. (x + half the linear coefficient)2 5. Solve by taking square roots. 6. Check your solutions. 2.4 Solving Equations Algebraically (Page 17 of 40) Example 4 Completing the Square Solve 2x 2 + 8x + 3 = 0 by completing the square. Example 5 Completing the Square Solve 3x 2 − 4x − 5 = 0 by completing the square. 2.4 Solving Equations Algebraically (Page 18 of 40) Derivation of the Quadratic Formula Solve ax 2 + bx + c = 0 by completing the square. The Quadratic Formula The solutions the quadratic equation ax 2 + bx + c = 0 are given by the Quadratic Formula −b ± b 2 − 4ac x= 2a Example 6 Solve by the Quadratic Formula Solve 3x 2 − 4x − 5 = 0 by using the quadratic formula. 2.4 Solving Equations Algebraically (Page 19 of 40) Example 7 No Real Solutions, No x-Intercepts 1. Solve 2x 2 − 3x + 5 = 0 . 2. Find the zeros and x-intercepts of f (x) = 2x 2 − 3x + 5 . Example 8 Two Real Solutions, Two x-Intercepts 1. Solve 2x 2 − 4x − 3 = 0 . 2. Find the zeros and x-intercepts of f (x) = 2x 2 − 4x − 3 . Example 9 One Real Solutions, One x-Intercepts 2 1. Solve 2x − 4x + 2 = 0 . 2. Find the zeros and x-intercepts of f (x) = 2x 2 − 4x + 2 . 2.4 Solving Equations Algebraically (Page 20 of 40) Example 10 Finding Room Dimensions A rectangular bedroom is three feet longer than it is wide and has an area of 154 square feet. Find the dimensions of the room. The Position Equation The height of an object above the earth’s surface is governed by the position equation, s = −16t 2 + v0t + s0 , where s = the height of the object in feet v0 = the initial velocity of the object s0 = the initial height of the object Example 11 Falling Time A construction worker on the 24th floor of a building project accidentally drops a wrench and yells “look out below!” Could a person at ground level hear the warning in time to get out of the way? The speed of sound is 110 feet per second. Assume each floor is 10 feet high. 2.4 Solving Equations Algebraically (Page 21 of 40) Example 12 Modeling Number of Internet Users From 2000 to 2007, the estimated number of Internet users I (in millions) in the United States can be modeled by the quadratic equation I = −1.163t 2 + 17.19t + 125.9 , 0≤t ≤7 where t is the number of years since 2000. Use the model to determine when the number of Internet users is predicted to be 180 million? Solve algebraically and graphically. Exercise 110 Dimensions of a Corral A rancher has 100 meters of fencing to enclose two identical, adjacent rectangular corrals (as diagramed). If the rancher wants to enclose a total area of 350 square meters, what are the dimensions of each corral? 2.5 Solving Other Types of Equations Algebraically (Page 22 of 40) 2.5 Solving Other Types of Equations Algebraically Polynomial Equations an x n + an−1 x n−1 ++ a2 x 2 + a1 x + a0 = 0 General Form: Linear: a1 x + a0 = 0 a2 x 2 + a1 x + a0 = 0 Quadratic: a3 x 3 + a2 x 2 + a1 x + a0 = 0 Cubic: Example 1 Solve Polynomial Equation by Factoring 3x 4 = 48x 2 Solve Example 2 Solve Polynomial Equation by Factoring 2x 3 − 6x 2 + 6x − 18 = 0 Solve Example 3 Solve Equation of the Quadratic Type x 4 − 3x 2 + 2 = 0 Solve 2.5 Solving Other Types of Equations Algebraically (Page 23 of 40) Example 4 Solve Equations With Radicals a. Solve 2x + 7 − x = 2 b. Solve 2x − 5 − x − 3 = 1 Example 5 Solve Solve an Equation with a Rational Exponent (x − 4)2/3 = 25 2.5 Solving Other Types of Equations Algebraically (Page 24 of 40) Example 6 Solve an Equation with Fractions 2 3 Solve = −1 x x−2 Solving Absolute Value Equations Let E be any expression and k be a real number, k ≥ 0 . If E = k , then E = k or E = −k For Example, if x = 4 , then x = 4 or x = - 4. Example 7 Solve Solve an Absolute Value Equation x 2 − 3x = − 4x + 6 2.5 Solving Other Types of Equations Algebraically (Page 25 of 40) Example 8 Reduced Rates A ski club chartered a bus for a ski trip at a cost of $480. To lower the cost of the bus per person, the club invited nonmembers to go along. After 5 nonmembers joined the trip, the fare per skier decreased by $4.80. How many club members went on the trip? Compound Interest Formula r⎞ ⎛ A = P ⎜ 1+ ⎟ ⎝ n⎠ nt A = Amount invested after t years. P = initial principal deposited r = annual interest rate n = number of compounding periods per year Example 9 Compound Interest When you were born, your grandparents deposited $5000 in longterm investments in which the interest was compounded quarterly. Today, on your 25th birthday, the value of the investment is $25,062.59. What is the annual interest rate for the investment? 1.8 Other Types of Inequalities (Page 26 of 40) 2.6 Solving Inequalities Algebraically and Graphically A number satisfies an inequality in one variable if substituting the number for the variable results in a true statement. Such numbers are called solutions to the inequality. The solution set to an inequality is the set of all solutions to the inequality. To solve an inequality means to find all the solutions to the inequality. Two inequalities are said to be equivalent if they have the same solutions set. Some example of linear inequalities in one variable follow: −3 4 − x < 3 , 3x + 2 ≤ 5 , −8(2 − 3x) > 7 , x≥5 7 Graphs of Solution Sets and Interval Notation Since the solution set to a linear inequality is generally an infinite set, we can express the solution set as an inequality, a graph, or as an interval using what is referred to as interval notation. In Words Inequality Numbers less than 4 Numbers less than or equal to 4 Numbers greater than 4 Numbers greater than or equal to 4 Numbers between 0 and 4 Numbers between 0 and 4, including 4 Numbers between 0 and 4, including 0 and 4 x<4 Graph 0 4 0 4 0 4 0 4 0<x<4 0 4 0<x≤4 0 4 0≤x≤4 0 4 x≤4 x>4 x≥4 Interval Notation 1.8 Other Types of Inequalities (Page 27 of 40) Properties of Inequalities Let a, b, c and d be real numbers or expressions. 1. Transitive Property a < b and b < c a<c 2. Addition of Inequalities a < b and c < d a+c <b+d 3. Addition Property a<b 4. Multiplication Property For c > 0, a < b ac < bc For c < 0, a < b ac > bc a+c <b+c There are corresponding inequalities for >, ≥ and ≤ inequalities. Example 1 Solving Linear Inequalities Solve each of the following. Express the solution set as an inequality, a graph, and in interval notation. 1. 8 − 2x > 11 2. −3(4 − x) ≥ 14 + x 3. 2[x − (2x + 3)] ≤ 7 − 2(5 − x) 1.8 Other Types of Inequalities (Page 28 of 40) Example 2 Solving Double Inequalities Solve each of the following. Express the solution set as an inequality, a graph, and in interval notation. 1. 17 < 5 − 3x ≤ 29 2. −3 ≤ 6x − 1 < 3 1.8 Other Types of Inequalities (Page 29 of 40) Absolute Value Property for Inequalities For an expression E and positive number k. E < k is equivalent to −k < E < k 1. E > k is equivalent to E > k or E < −k 2. Example 3 Solving Absolute Value Inequalities Solve without a calculator. Describe the solution set as an inequality, in a graph and in interval notation. Verify with a graphing calculator. x ≤5 1. 2. x >5 Example 4 Solving Absolute Value Inequalities Solve 2x − 3 ≤ 9 without a calculator. Describe the solution set as an inequality, in a graph and in interval notation. Verify with a graphing calculator. 1.8 Other Types of Inequalities (Page 30 of 40) Example 5 Solving Absolute Value Inequalities Solve without a calculator. Describe the solution set as an inequality, in a graph and in interval notation. Verify with a graphing calculator. 3t + 4 > 12 1. 2. 7− x+2 ≥ 3 1.8 Other Types of Inequalities (Page 31 of 40) Example 6 Solve a Polynomial Inequality x 2 − 2x < 3 Solve Fact: A polynomial can only change signs at its zeros (the values of x that make the polynomial equal to zero). These zeros are the critical numbers of the inequality, which divide the number line into test intervals for the inequality. STEP 1 Write the inequality so that one side is a polynomial and the other side is zero. STEP 2 Find all the real zeros of the polynomial, that is, the critical numbers of the inequality. STEP 3 Arrange the critical numbers along a “working number line” showing the test intervals. STEP 4 Evaluate the polynomial at a test value in each test interval. The polynomial will have the same sign for every value on each test interval. 1.8 Other Types of Inequalities (Page 32 of 40) Example 7 Solving a Polynomial Inequality 1. Solve x 2 − x < 6 2. Solve 2x 3 − 3x 2 − 32x > −48 1.8 Other Types of Inequalities (Page 33 of 40) Example 8 Unusual Solution Sets a. Solve x 2 + 2x + 4 > 0 b. Solve x 2 + 2x + 1 ≤ 0 c. Solve x 2 + 3x + 5 < 0 d. Solve x 2 − 4x + 4 > 0 1.8 Other Types of Inequalities (Page 34 of 40) Example 9 Solving a Rational Inequality 2x − 7 Solve ≤ 3. x−5 Fact: A rational expression can only change signs at the zeros of the numerator and denominator. These zeros are the critical numbers of the inequality, which divide the number line into test intervals for the inequality. Example 10 Finding the Domain Find the domain of 64 − 4x 2 Recall: The domain of an expression is the set of all x-values so that the expression is a real number. 1.8 Other Types of Inequalities (Page 35 of 40) Example 11 Comparative Shopping You need to choose between two different cell phone plans. Plan A costs $49.99 per month for 500 minutes plus $0.40 for each additional minute. Plan B costs $45.99 per month for 500 minutes and $0.45 for each additional minute. Beyond the 500 included minutes, how many additional minutes must you use for plan B to cost more than plan A? Example 12 Height of a Projectile A projectile is fired straight upward from ground level with an initial velocity of 384 feet per second. During what time period will its height exceed 2000 feet? Recall: s(t) = −16t 2 + v0 t + s0 . 1.8 Other Types of Inequalities (Page 36 of 40) 2.7 Linear Models and Scatter Plots Correlation Example 1 Scatter Plots The data in the table shows the outstanding household credit market debt D (in trillions of dollars) from 1998 through 2004. a. Construct a scatter plot of the data. Year 1998 1999 2000 2001 2002 2003 2004 b. Household credit market debt, D (trillions of dollars) 6.0 6.4 7.0 7.6 804 9.2 10.3 Find the linear regression equation that models D. 1.8 Other Types of Inequalities (Page 37 of 40) Linear Regression Function The linear regression line is the line that mathematically best fits the data. The linear regression equation, or linear regression function, is the equation of the regression line. Finding the Linear Regression Equation on the TI-83 1. Enter the independent variable data values into list 1 (L1) and the corresponding dependent variable values into list 2 (L2). To access your lists press STAT followed by ENTER. STAT / 1:Edit 2. Press STAT PLOT followed by ENTER. Then set the Plot1 settings as shown. Press ZOOM / 9:ZoomStat to view the scattergram. 3. Run the linear regression program: STAT / CALC / 4:LinReg (ax+b) Lx , Ly , Y1 On the TI-83 a is the slope and (0, b) is the “y”-intercept (i.e. the vertical axis intercept). Write a and b to three decimal places. 3. Rewrite the equation using the variables in the application. 1.8 Other Types of Inequalities (Page 38 of 40) Example 2 The American life span has been increasing over the last century. Let L(t) represent the life expectancy at birth for Birth Life an American born t years after 1970. Year Expectancy 1. Create a scatter plot of the data on your 1970 70.8 1975 72.6 calculator and determine if a linear 1980 73.7 model is appropriate. If it is, then find 1985 74.7 the linear regression model for the data. 1990 1995 2000 75.4 76.1 76.9 2. Use the model to predict the life expectancy for someone born in 2010. 3. Use the model to predict the birth year for someone whose life expectancy at birth is 66 years. 4. Interpret the meaning of the slope in this application. 1.8 Other Types of Inequalities (Page 39 of 40) Example 3 Let P represent the salmon population (in millions) at t years since 1950. 1. Create a scatter plot of the data on your calculator and determine if a linear model is appropriate. If it is, then find the linear regression model for the data. 2. Predict the salmon population in year 1955. Number of Years since 1950 t 10 15 20 25 30 35 40 Salmon Population (millions) P 10.02 10.00 7.61 3.15 4.59 3.11 2.22 3. Predict the year when the salmon population was 6.4 million. 4. Explain the meaning of the slope in this situation. 1.8 Other Types of Inequalities (Page 40 of 40) Example 4 The table shows the average salaries for professors at four-year colleges and universities. Let s represent the average salary (in thousands of dollars) at t years since 1970. a. Verify the linear regression equation for s is s = 1.70t + 6.71 . b. Predict the average salary in 2008. c. Year Average Salary (thousands of dollars) 1975 1980 1985 1990 1995 2000 16.6 22.1 31.2 41.9 49.1 57.7 Predict when the average salary will be $75,000. d. Explain the meaning of the slope in this situation.